An even-degree polynomial-in-OU volatility model with negative spot-vol correlation prices its forward below spot. Sotnikov and Viot establish this across the family of exponentially fading memory (EFM) signature volatility models. Polynomial-in-OU models belong to that family, including the Quintic (degree 5, and therefore admissible). Neither mean reversion nor a stationary start shifts the boundary.
The paper studies a generic asset under the pricing measure. The SPY application discussed below is ours.
Why does the forward fall below spot?
The stock dynamics under the pricing measure are dS = SΣdB. This stochastic exponential is always a local martingale and a supermartingale, giving E[S_T] ≤ S_0. A strict inequality means the model's value for a forward on S_T differs from S_0. Put-call parity then fails within the model.
The defect offers no trade. It leaves the model with prices that cannot all be consistent.
Building volatility from fading memory
A path signature collects all the iterated integrals of a path. The path here is Ŷ_t = (t, W_t, Z^1_t,..., Z^d_t), with 2 + d channels. W has correlation ρ ∈ [−1, 1] with the price Brownian motion; the Z factors are independent. Volatility comes from a linear functional of the signature, truncated at order N. It behaves like a degree N polynomial in path features. The coefficient σ^{2⊗N}, attached to W_t^N/N!, serves as its leading coefficient.
EFM discounts each iterated integral using the sum of its letters' decay rates λ. Older shocks therefore carry less weight. As the authors explain, "finite linear functions of the classical signature do not, in general, yield a nontrivial stationary process". When every λ_i > 0, the EFM signature admits a stationary law by starting at minus infinity.
The model adds two controls. A random initial signature Ξ records the path before time 0. Choosing its stationary law makes volatility stationary from the outset, provided g is constant. The other control, g(t), scales volatility to the initial forward variance curve and lies between g_* > 0 and g^*. The Quintic corresponds to N = 5, α_2 = α_4 = 0, λ_2 > 0, Ξ = ∅.
Theorem 3.3 requires a nonzero coefficient on W_t^N/N!. The price is a true martingale when N = 1 or ρ = 0. For N ≥ 2 and ρ ≠ 0, true martingality holds if and only if N is odd and ρσ^{2⊗N} < 0.
The proof turns on explosion. Sin's change of measure makes martingality equivalent to non-explosion of an SDE. Its drift is ρ times the volatility functional, applied to the signature of the SDE's own solution. The authors follow U_t, a fading integral of that process, which explodes at the same time. At large U, its drift behaves like ρg(t)σ^{2⊗N}U^N/N!. An odd N and a negative product pull U back. Otherwise, U grows like the ODE x' = cx^N. Moving from n to n + 1 takes at most Cn^(−N); for N ≥ 2, these times have a finite sum. U consequently blows up before T with positive probability.
The old martingale gate still applies
Abi Jaber, Gassiat and Sotnikov established the classical criterion with λ = 0, Ξ = ∅, d = 0, g ≡ 1. Their result also covered a multidimensional version with d > 0 and ρ = 1. For the Quintic, Abi Jaber, Illand and Li proved sufficiency alone under α_5 > 0 and ρ ≤ 0. Sotnikov and Viot obtain the full if-and-only-if at any degree while allowing fading memory, random initial memory and forward-variance scaling together.
Only three inputs determine the criterion: N, the sign of ρ and the sign of σ^{2⊗N}. It ignores lower-order coefficients. Coefficients attached to words containing Z letters also leave it unchanged. Once ρ ≠ 0, every even N fails. The product's sign determines whether U explodes toward +∞ or −∞.
For equity-index modelling, the restriction is light. Negative ρ and a positive leading coefficient are already the configuration assumed by the Quintic sufficiency result. The restriction matters when calibration selects an even degree or gives the leading coefficient the same sign as ρ.
Where the reader must rely on earlier proofs
Four proofs are omitted. The authors say three follow the classical-case arguments: the first assertion of Lemma 3.6, Lemma 4.1, and Step 2 of the general-d necessity proof. They describe the fourth, Lemma 4.2, as a slight variation of a Revuz and Yor exponential inequality. Step 2 establishes that the blow-up event has positive probability. Necessity for d > 0 therefore depends on an adaptation the reader is asked to accept. For the two-sided construction supporting the stationary start, the paper refers to Abi Jaber and Sotnikov (2025).
A parameter choice can also break an assumption before the theorem applies. When Ξ = ∅, the signature at t = 0 has only its constant entry. The forward-variance normaliser at t = 0 is consequently α_0². Choosing α_0 = 0 makes g(0) infinite and violates the theorem's requirement that g remain below g^*.
Before putting SPY options through a backtest
For a calibrated parameter set, the theorem gives four checks:
- N odd, with σ^{2⊗N} ≠ 0;
- σ^{2⊗N} opposite in sign to ρ;
- g bounded on [0, T], requiring α_0 ≠ 0 when Ξ = ∅ and forward-variance scaling is used;
- Ξ having finite moments of every order, as the stationary choice does.
The paper gives a desk no evidence for preferring this model. It contains proofs throughout, with no calibration, pricing or hedging experiment. Whether it improves calibration accuracy against the Quintic remains open, as do fit stability, transaction costs and out-of-sample hedging performance. Even the necessity result is qualitative: explosion occurs with positive probability before T, without a numerical bound on that probability.
Our work begins at that gap. The paper's generic continuous-time asset under the pricing measure is something we cannot trade directly. We instead instantiate the model on SPY and its options and hedge at daily intervals. Such a hedge tests our application of the model. Its results say nothing about the theorem itself. Nor can finite Monte Carlo certify true martingality: a positive-probability explosion may occur on paths the simulation never samples. We therefore impose the odd-degree and sign conditions analytically before admitting parameters to simulation. The evaluation is still running. We have no figures to report yet.
The paper belongs in model validation, before a backtest. Its extension to stationary volatility and random initial memory passes the same filter as the classical model, and applying that filter is cheap. A calibration showing that the stationary start fits the SPX term structure better than the Quintic with Ξ = ∅ would change my view. The paper conducts no such test.